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440
AIAA
J O U R N A L
V O L .
23, NO. 3
Nonlinear Analysis of Laminated Shells
Including Transverse Shear Strains
J . N. R eddy* and K. Chandrashekharat
Virginia
Polytechnic
Institute
a nd State University,
Blacksburg, Virginia
Abstract
N
UMERICAL results obta ined us ing a doubly curved,
shear deformable shell element are presented for
geometr ical ly nonlinear anlaysis of laminated composite
shells. The element is based on an extension of Sanders ' shell
theory and accounts for the von Karman strains and trans-
verse
shear strains. The sample numerical results presented
here
for the
geometr ica lly nonlinear analys is
o f
laminated
composite shells should serve
as
references
fo r
fu tu re
in -
vestigations.
Contents
Laminated shells are f inding increased applications in
aerospace, automobile, and petrochemical industries. This is
due
primarily to the high stiffness-to-weight
ratio,
high-
strength-to-weight
ra t io , and lower machining and main-
tenance costs associated with composite structures. However,
the analysis of com posite structures is more com plicated when
compared to meta ll ic s tructures , because laminated composite
structures
are anisotropic and characterized
by
bending-
stretching coup ling. Fu rther, the classical shell theories, whic h
ar e based on the
K i rc h h of f -Love
kinematic hypothesis, are
k n o w n to y ield deflec tions and stresses in laminated shells that
are as much as
3 0%
in error. This error is due to the neglect of
t ransverse shear strains in the classical shell theories.
T he effects o f
t ransverse shear deformat ion
an d
thermal
expansion through
th e
th ickness
fo r
laminated, t ransversely
isotropic, cylindrical shells were considered by Zukas and
Vinson.
1
Dong an d
T so
2
constructed a laminated o rthotropic
shell
theory that includes transverse shear deformation. Other
shear deformation theories, specialized to anisotropic cylin-
drical shells, were presented by Whitney and
Sun .
3
'
4
Recent ly ,
Reddy
5
extended
Sanders '
theory to account for the trans-
verse
shear strains, and presented exact solutions for simply
supported cross-ply laminated shells. All of these studies are
limited to
small displacement theories
a nd
static analyses.
In the present paper , the extended Sanders ' shell theory that
accounts for the shear deformation and the von Karman
strains is used to develop a displacement
finite
element model
fo r
the bending analysis of laminated composite shells.
Numer ica l
results for cylindrical and doubly curved shells are
presented, showing the effect of geometry and material or-
thotropy
on the
def lect ions
an d
stresses.
Consider a laminated shell constructed of a finite n u m b e r
of
uniform -thickness orthotropic layers, or iented arbit rari ly
with
respect
to the
shell coordinates ( ?
7
, b * f ) .
The o r -
thogonal cu rviline ar coordin ate system £ / , £ 2 > D is chosen
such
that the £ / and £
2
curves are lines of curvature on the
midsurface f = 0 , and f curves are straight lines perpendicular
to the
surface
f = 0 .
Received N ov.
29 ,
1983; revision received
M a r c h 1 5,
1984.
Copyr igh t
© Amer i can
Inst i tute
of Aeronaut i cs and As tronaut i cs ,
Inc., 1984.
All
rights
reserved. Ful l paper to be avai lable
from
National Technical Information
Service,
Springfield, Va. ,
at the
s tandard
pr ice.
*Professor ,
D e p a r t m e n t o f
Engineering Science
an d
Mechanics.
jGraduate
Research
A ssistant ,
D epar tment
of Engineer ing Science
an d
M echanics.
The principle of virtual displacements can be used to derive
th e
governing equat ions
5
'
6
:
dN,
i
x
dN
2
—l
dx
2
.dx,
d x
2
d
X ]
where
c
0
=
0.5 \/R
2
- 1
/ / ? / ) ,
and
du
3
dx?
(1 )
T—
dx
2
du
3
du
3
—t
dx
2
(2)
The rel ationship between the stress resultants T V / , M /, Q ,) are
generalized displacements (u , v,
w ,
< / > / , 4 >
2
) can be found in
R e f .
6.
The finite elem ent model of the equation s is of the form
where [A) = ( \ u
l
) , \u
2
} ,
(u
3
J ,
{ < / > / ,
4 >
2
} )
7
, [A] is the
element
stiffness
matrix, and
( F )
is the force vector. The
coefficients
of the stiffness matr ices ar e included in R e f . 6.
Note that th e stiffness matr ix [K ] is a
funct ion
of the
u n k n o w n solution vector ( A ) ; therefore, an iterative solution
procedure is required for each
load
step. In the present study,
the authors have used the direct (i.e., Picard-type) iteration
technique. Sample numerical results for laminated shell
problems are presented in Figs. 1-3. A discussion of these
results
i s
presented
in the
fo llowing paragraphs.
The following types of boundary conditions for a quarter-
panel were used (along the symmetry lines, normal in-plane
displacement , an d normal rotation are set to zero):
= w=(f>
J
=0 at y —
= w = l >
2
= 0 at x=a
4)
Here
u, v,
an d
w
denote th e displacements along th e three
coordinate axes, an d
< >
7
an d < />
2
denote the rotations of the
transverse normals
abou t
the x
2
an d x
I
axes, respectively. T he
fol lowing material properties
ar e
used:
Material 1:
E, =25 x
10
6
psi, E
2
= 10
6
psi,
G
]2
=
G
]3
=0.5 x
10
6
psi
G
23
=
0.2x
10
6
psi, v,
2
= 0.25 (5a)
Materia l
2:
Ej =
40 x
10
6
psi,
E
2
=
10
6
psi,
G
]2
= G
}3
=
0.6 x
10
6
psi
G
23
=0.5 x 10
6
psi,
V] 2
=0 .25 (5b)
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MARCH 1985
NONLINEAR
A N A L Y S I S
OF LAMINATED SHELLS
441
Load
P
= F
3 .
0.0 0.2 0.4 0.6 0.
Deflection, -w/h)
Fig.
1 Bending of a nine layer cross ply [0°/90°/0 ..] s imply
supported
spherical shell
subjected t o a uniformly distributed load.
cross-ply
I
Deflection, -w
10
in)
• — angle-ply
0.0 0.1 0.2 0.3 0.4 0.5
3.0 i———i———i———i———i———i———i———i———i———i———|
angle-oly
Load, 2.0
q
n
psi)
0.4
Load ,
ci
nsi)
0.1 0. 2 0.3 0.4
Deflection, w
in in.)
0.0
.5 -»-cross-ply
Fig. 2
Bending
of a two layer cross ply and angle ply s imply sup
ported spherical shells under uniform
loading.
T he
first
problem is concerned with the bending of a nine-
layer [0
0
/90°/0°/90%..] cross-ply spherical
panel
under
external pressure load. T he following geometrical data ar e
used
in the
analysis: RI
=R
2
=R= 1000 in., a = b =5 Q in.,
h =
1 in. Individual layers are assumed to b e of equal thickness
h j = h/9). Results using
tw o
sets
of
orthotropic-material
constants, typical
of
high-modulus graphite-epoxy material
(the
ratios are more pertinent here) for in d iv idua l layers, are
presented
i n Fig . 1.
Figure 2 contains the pertinent data an d results (with
different
scales)
fo r
two-layer cross-ply
an d
angle-ply
simply
supported spherical panels (of material 2). It is interesting to
note that the
type
o f
nonlinearity
exhibited by the two
shells
i s
quite different;
th e
cross-ply
shell
gets
softer,
whereas
th e
arigle-ply
shell gets stiffer with an increase in the applied load.
While both
shells have bending-stretching coupling due to the
laminat ion
scheme (B
22
=
— B
11
nonzero for the cross-ply
shell an d
B
} 6
an d B
2
6 ar e nonzero for the angle-ply shell), the
Fig.
3
Bending
of a
c lamped
cross ply
[ 0 ° / 9 0 ° ] cylindrical shell
under
a
uniform loading.
angle-ply
experiences shear coupling that stiffens the spherical
shell
relatively more than the normal coupling (note that, in
general, shells get
softer
under
externally applied pressure
loads).
Lastly,
Fig.
3 contains
results (i.e.,
w ,
o
y
, o
xz
vs
load)
for a
cross-ply [0°/90°] clamped cylindrica l
panel
under un i form
load. Clearly, the
load
deflection curve indicates
that
th e
nonlinearity exhibited by the cross-ply shell is of the har-
dening
type.
The shear-flexible finite
element
presented here is com-
putat ional ly more efficient than those based on fully o r
degenerated three-dimensional elasticity theory. Since only
the von Karman-type nonlinearity is accounted for here, the
element is not suitable for
applicat ions involving
severe
dis tort ions (i.e., large strains). F rom the numerica l com-
putat ions it is
observed
that
boundary condi t ions
on the in-
plane d isplacements
have
a significant effect on the shell
deflections and stresses. It is
also
observed
that
the
form
of
nonlinearities exhibited by laminated shells depends on the
lamination schemes
(i.e., cross-ply
a nd
angle-ply).
It
wou ld
be
of considerable interest
to
verify these
f indings by ex-
periments .
Acknowledgments
T he
results reported
here were obtained dur ing in -
ves t igations supported by Structura l Mechanics Divisions of
AFOSR and NASA
Lewis .
The authors are
grateful
to Dr.
Anthony A mo s and Dr. Chris Chamis for the support .
References
Zukas , J . A. and Vinson, J. R., "Laminated
Transversely
Isotropic
Cylindrical
Shells, Journa l
of
Applied Mechanics, Vol .
38 ,
J u n e
1971,
pp .
400-407.
2
D o n g , S. B. and Tso,
F .K .W . ,
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